Don't double up the yarn, it will make the heel too bulky and uncomfortable in your shoes. You can go pick up some nylon thread at a craft store, or check your LYS for reinforcing thread. Just knit it with your sock yarn holding the two strands together.
Another option is to knit the heel flap using a K1, S1 pattern. It bulks up the heel just a bit so that it's sturdier, but you don't have to use a separate strand. Check out my sock knitting tutorial for this technique.